Representations of the symmetric group in deformations of the free lie algebra
We consider, for a given complex parameter a, the nonassociative product defined on the tensor algebra of á-dimensional complex vector space by the left-normed bracketing is defined recursively to be the bracketing sequence The linear subspace spanned by all multilinear left-normed bracketings of homogeneous degree n, in the basis vectors is then an 5M-module Vn(a). Note that Vn(l) is the Lie representation Lie. of S. afforded by the áth-degree multilinear component of the free Lie algebra. Also, K.(-l) is the subspace of simple Jordan products in the free associative algebra as studied by Robbins [Ro]. Among our preliminary results is the observation that when a is not a root of unity, the module V.(a) is simply the regular representation. Thrall [T] showed that the regular representation of the symmetric group S. can be written as a direct sum of tensor products of symmetrised Lie modules Vi. In this paper we determine the structure of the representations V.(a) as a sum of a subset of these Vx. The Vx, indexed by the partitions X of n, are defined as follows: let m! be the multiplicity of the part i in X, let Lie, be the Lie representation of 5, and let ik denote the trivial character of the symmetric group denote the character of the wreath product Sm.[Si] of Smiacting on copies of. Then Vxis isomorphic to the Our theorem now states that when a is a primitive pi root of unity, the module V.(a) is isomorphic to the direct sum where X runs. © 1994 American Mathematical Society.
Calderbank, AR; Hanlon, P; Sundaram, S
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